Slim exceptional sets in Waring's problem: One square and five cubes

Amongst the most irritating problems of Waring type is that of establishing the expected asymptotic formula for the number of representations of an integer as the sum of five cubes and a square of natural numbers, The technology currently avasilable to practitioners of the Hardy-Littlewood method fails, by only the narrowest of margins, to deliver the sought-after conclusion, and indeed Vaughan [5] has succeeded in establishing a lower bound for the desired number of representations that misses that expected by only a positivs constant factor. Then purpose of this note is to demonstrate that, although the expected asymptotic formula may occaisionally fail to hold, the set of sich exceptional instances is extremely sparse.